To properly demonstrate the advantages of the Bootstrap DPSD estimator, the statistical performance under different conditions must be analyzed.
First of all, the methodology is explained thoroughly, from the simulation of dual-polarized weather radar signals to the computation of the statistical errors.
Also, the relationship between the bulk and spectral polarimetric radar variables is studied.
Next, the ideal DPSD estimator is analyzed in terms of statistical errors and their dependence with relevant parameters.
Then, the analysis of the performance of the Bootstrap DPSD estimator is presented in three different categories: single signal single dwell, single signal multiple dwells, and dual signal.
Finally, an overview of the error performance of the Bootstrap DPSD estimator is presented at the end of the chapter.

\section{Methodology}
\subsection{Synthetic I/Q Time-Series Weather-Signal Simulator}
A dual polarimetric version of the weather-like signal simulator \citep{Zrnic1975a} is used, as presented in \citet{Torres2003a} and \citet{Yu2012}.
Let $V_\mathrm{H}(m)$ and $V_\mathrm{V}(m)$ be the I/Q time-series signals of the H and V channels, respectively.
As shown in \citet{Torres2003a}, and by following the procedure in \citet{Galati1995}, these can be obtained as:
\be \begin{aligned} V_\mathrm{H} (m)& = \sqrt{S_\mathrm{H}} X(m) + N_\mathrm{H}(m),\\
V_\mathrm{V} (m)& = \sqrt{S_\mathrm{V}} \Big[ \rho_\mathrm{HV} X(m) + \sqrt{1-\rho_\mathrm{HV}^2} Y(m) \Big] e^{-j\phi_\mathrm{DP}} + N_\mathrm{V}(m) , \end{aligned} \label{eq:simV} \ee
where $S_\mathrm{H}$, $S_\mathrm{V}$, $\rho_\mathrm{HV}$ and $\phi_\mathrm{DP}$ are the true values of the bulk radar variables, $N_\mathrm{H}(m)$ and $N_\mathrm{V}(m)$ are additive white Gaussian noise (AWGN), and $X(m)$ and $Y(m)$ are unit power, independent time-series signals obtained as \citep{Zrnic1975a}:
\be X(m) = \mathcal{F}^{-1} \{  \sqrt{-sS(k) \ln{u(k)} }  e^{j\theta(k)}  \},  \label{eq:xmzrnic} \ee
where $sS(k)$ is the Gaussian PSD model of the weather signal, $u(k)$ is an independent and identically distributed (IID) random variable with uniform distribution between 0 and 1, and $\theta(k)$ is also an IID random variable but with uniform distribution between $-\pi$ and $\pi$.
$Y(m)$ is obtained similarly from (\ref{eq:xmzrnic}) but using independently generated $u(k)$ and $\theta(k)$.
With this procedure, the signal parameters that can be modified are: mean radial velocity ($\bar{v}_\mathrm{r}$) and spectrum width ($\sigma_v$) of the Gaussian PSD model, the bulk radar variables in terms of signal power ($S_\mathrm{H}$ or $S_\mathrm{V}$), signal-to-noise ratio ($\mathrm{SNR}_\mathrm{H}$ or $\mathrm{SNR}_\mathrm{V}$), differential reflectivity ($Z_\mathrm{DR} = S_\mathrm{H}/S_\mathrm{V}$),  correlation coefficient ($\rho_\mathrm{HV}$) and differential phase ($\phi_\mathrm{DP}$).
Additionally, radar acquisition parameters can also be modified, which include maximum unambiguous velocity ($v_\mathrm{a}$) and number of samples per dwell ($M$). 
Multiple realizations can be produced to get $K$ independent spectra, and statistical properties of the estimator are computed running $N$ iterations in the simulation.
With (\ref{eq:simV}), the coefficients of the DPSDs corresponding to the signal follow a constant value which agrees with the true value of the bulk polarimetric radar variable.
However, the DPSDs of weather radar signals are not necessarily constant for all radial velocities.
To simulate different profiles of DPSDs as a function of radial velocity, the method described in \citet{Yu2012} is more appropriate, as it allows the DPSDs to follow a model for each spectral coefficient.
Physically, the presence of turbulent winds may produce signals with constant spectral polarimetric variables \citep{Yanovsky2011}.
%The physical interpretation of a signal with constant $sZ_\mathrm{DR}$ is that the radar resolution volume consists mostly of scatterers with approximately the same electrical size during the observation time, while a constant $s\rho_\mathrm{HV}$ indicates that the scatterers have similar homogeneity properties.
Nevertheless, the assumption that the DPSDs are constant in the spectrum is sufficient to analyze the statistical performance of the DPSD estimates.
This type of synthetic simulation allows generation of virtually any desired signal or composite signal under different scenarios, making it a powerful tool to study the statistical properties of any estimator.
However, the challenge in simulating a realistic signal using this method is that while it is possible to modify the simulation parameters to match actual observations, the mechanisms that drive the underlying phenomena may not be emulated.
Further discussion on this type of physical simulators is presented in Chapter \ref{chap_5}.
Throughout this chapter, the signal parameters were selected to resemble typical observations. 
The SNR is arbitrarily set high (20 dB) such that the noise contamination is minimal.
While a Gaussian assumption for the simulated signals may not be the most realistic case, it provides a simple model that can capture features of the signal such that quantitative performance analyses can be conducted.
% Fill in as necessary.

%since the signals of interest are those of hydrometeors (of which the polarimetric characteristics are better understood) and  though for smaller raindrops, it remains mostly constant along the spectrum width.

\subsection{Statistical Error Calculation}
Assuming the signal originates from a single group of uniform scatterers, for each set of independent estimates ($s\hat{Z}_\mathrm{DR}$ and $s\hat{\rho}_\mathrm{HV}$), the spectral coefficients above a SNR threshold are used to compute the average spectral errors. % for which $s\hat{\mathrm{SNR}}_\mathrm{H} > 20$ dB and $s\hat{\mathrm{SNR}}_\mathrm{V} > 20$ dB are averaged.
Herein, the bulk radar variables are used as the true value to compute the errors because the signal is modeled as such.
If we used a more complex model as described above, the errors would become a function of the spectral coefficient and the statistical analysis would unnecessarily become more complicated.
The average value of the SNR-thresholded spectral coefficients represents the strong signal components with minimal noise.
These are obtained as
\be \overline{s\hat{Z}_\mathrm{dr}} = \langle s\hat{Z}_\mathrm{dr}(k') \rangle , \ee
\be \overline{s\hat{\rho}_\mathrm{HV}} = \langle s\hat{\rho}_\mathrm{HV}(k') \rangle ,\ee
where $k'$ are the spectral coefficients for which $s\hat{\mathrm{SNR}}_\mathrm{H}$ and $s\hat{\mathrm{SNR}}_\mathrm{V}$ are greater than 20 dB, and $\langle \cdot \rangle$ indicates averaging in the frequency domain.
%for $k' = k | s\hat{\mathrm{SNR}}_\mathrm{H}(k) > 20$ dB and $s\hat{\mathrm{SNR}}_\mathrm{V}(k) > 20$ dB.
The bias of $s\hat{Z}_\mathrm{DR}$ is then obtained as
\be \mathrm{bias}(s\hat{Z}_\mathrm{dr}) = E\bigg[{\overline{s\hat{Z}_\mathrm{dr}}} - Z_\mathrm{dr} \bigg] \simeq \frac{1}{N}\sum \limits ^{N-1} _{n=0} \bigg( {\overline{s\hat{Z}_\mathrm{dr}}}_{,n} - Z_\mathrm{dr} \bigg) , \label{eq:error_1}\ee
\be \mathrm{bias}(s\hat{Z}_\mathrm{DR}) \simeq 10 \log \left[ 1 + \frac{\mathrm{bias}(s\hat{Z}_\mathrm{dr})}{Z_\mathrm{dr}} \right]  (\mathrm{dB}), \ee
where $N$ is the number of independent DPSDs used to study the statistical variability of the spectral estimates, not to be confused with the $K$ independent spectra used to compute the DPSDs.
The standard deviation (SD) of $s\hat{Z}_\mathrm{DR}$ is computed as
\be \mathrm{SD}(s\hat{Z}_\mathrm{dr}) = \sqrt{E\bigg[ \bigg({\overline{s\hat{Z}_\mathrm{dr}}} - Z_\mathrm{dr} \bigg)^2 \bigg]} \simeq \sqrt{\frac{1}{N} \sum \limits ^{N-1} _{n=0} \bigg( {\overline{s\hat{Z}_\mathrm{dr}}}_{,n} - Z_\mathrm{dr} \bigg)^2} , \ee
\be \mathrm{SD}(s\hat{Z}_\mathrm{DR}) \simeq 10 \log_{10} \left[ 1 + \frac{\mathrm{SD}(s\hat{Z}_\mathrm{dr})}{Z_\mathrm{dr}} \right] (\mathrm{dB}). \ee
The normalized bias and SD of $s\hat{\rho}_\mathrm{HV}$ are
\be \frac{\mathrm{bias}(s\hat{\rho}_\mathrm{HV})}{\rho_\mathrm{HV}} =   \frac{E\bigg[{\overline{s\hat{\rho}_\mathrm{HV}}} - \rho_\mathrm{HV} \bigg]}{\rho_\mathrm{HV}} \simeq \frac{\frac{1}{N}\sum \limits ^{N-1} _{n=0} \bigg( {\overline{s\hat{\rho}_\mathrm{HV}}}_{,n} - \rho_\mathrm{HV} \bigg) }{\rho_\mathrm{HV}} , \ee
\be \frac{\mathrm{SD}(s\hat{\rho}_\mathrm{HV})}{\rho_\mathrm{HV}} = \frac{\sqrt{E\bigg[ \bigg({\overline{s\hat{\rho}_\mathrm{HV}}} - \rho_\mathrm{HV} \bigg)^2 \bigg]}}{\rho_\mathrm{HV}} \simeq \frac{\sqrt{\frac{1}{N} \sum \limits ^{N-1} _{n=0} \bigg( {\overline{s\hat{\rho}_\mathrm{HV}}}_{,n} - \rho_\mathrm{HV} \bigg)^2}}{\rho_\mathrm{HV}} . \label{eq:error_6}\ee
Equations (\ref{eq:error_1})-(\ref{eq:error_6}) will be used later to quantify the dependence on the different parameters under analysis.

\subsection{Relationship between Bulk and Spectral Radar Variables}
As presented by \citet{Yu2012}, the values of the bulk polarimetric variables may be obtained from the spectra as:
\be Z_\mathrm{dr} = \frac{1}{P_\mathrm{V}} \int \limits ^\infty _{-\infty} sZ_\mathrm{dr}(f) sS_\mathrm{V}(f) df , \ee
\be \rho_\mathrm{HV} = \frac{1}{\sqrt{P_\mathrm{H} P_\mathrm{V} }} \left| \int \limits ^\infty _{-\infty} sS_\mathrm{X}(f) df\right| , \ee
where $sS_\mathrm{X}$ is related to $s\rho_\mathrm{HV}$ by 
\be s\rho_\mathrm{HV}(f) = \frac{|sS_\mathrm{X}(f)|}{\sqrt{sS_\mathrm{H}(f) sS_\mathrm{V}(f)}} .\ee
In other words, the bulk variables are obtained by integrating power-weighted DPSDs.
While bulk variables provide a good insight about the underlying phenomena for single signals, it is clear that they can be biased when there is more than one signal in the spectra.
The spectral polarimetric variables add another dimension (of velocity) to the bulk polarimetric variables, and display additional information about the scatterers in a resolution volume that is not depicted in bulk variables.
%Furthermore, by applying a SNR threshold on the spectral coefficients and averaging them, the noisy spectral coefficients are disregarded, which can potentially improve the quality of the estimates. %though not analyzed in this work
Furthermore, the average of the thresholded coefficients is arguably a better way to obtain bulk estimates, provided that the statistical error of the DPSDs are better than those of bulk estimates, since the noise spectral coefficients can be ignored altogether.

Additionally, the errors for bulk and spectral variables have a different parameter dependence.
An extensive analysis on the errors of bulk polarimetric variables was done by \citet{Melnikov2004} and the analytical expressions for the errors were obtained with the perturbation method as:
\be \mathrm{bias}(\hat{Z}_\mathrm{DR}) = \frac{10}{M \ln{10}} \left[ \frac{1+2\mathrm{SNR}_\mathrm{V}}{\mathrm{SNR}_\mathrm{V}^2} + \frac{0.56(1-\rho_\mathrm{HV}^2)}{\sigma_{vn}}\right] ,\label{eq:bulk_1}\ee
\be \mathrm{SD}(\hat{Z}_\mathrm{DR}) = \frac{10}{\sqrt{M} \ln{10}} \sqrt{ \frac{1+2\mathrm{SNR}_\mathrm{H}}{\mathrm{SNR}_\mathrm{H}^2} +  \frac{1+2\mathrm{SNR}_\mathrm{V}}{\mathrm{SNR}_\mathrm{V}^2} + \frac{1.13(1-\rho^2_\mathrm{HV})}{\sigma_{vn}} } ,\label{eq:bulk_2}\ee
\be \frac{\mathrm{bias}(\hat{\rho}_\mathrm{HV})}{\rho_\mathrm{HV}} = \frac{1}{M} \left[ \frac{2\mathrm{SNR}_\mathrm{H}+3}{8\mathrm{SNR}_\mathrm{H}^2} + \frac{2\mathrm{SNR}_\mathrm{V}+3}{8\mathrm{SNR}_\mathrm{V}^2} + \frac{\mathrm{SNR}_\mathrm{H} + \mathrm{SNR}_\mathrm{V} + 1}{4\mathrm{SNR}_\mathrm{H}\mathrm{SNR}_\mathrm{V}\rho^2_\mathrm{HV}} + \frac{0.14(1-\rho^2_\mathrm{HV})^2}{\sigma_{vn}\rho^2_\mathrm{HV}} \right] ,\label{eq:bulk_3}\ee
\be \frac{\mathrm{SD}(\hat{\rho}_\mathrm{HV})}{\rho_\mathrm{HV}} = \sqrt{ \frac{1}{M} \left[ \frac{1-2\mathrm{SNR}_\mathrm{H}}{4\mathrm{SNR}_\mathrm{H}^2} + \frac{1-2\mathrm{SNR}_\mathrm{V}}{4\mathrm{SNR}_\mathrm{V}^2} + \frac{\mathrm{SNR}_\mathrm{H} + \mathrm{SNR}_\mathrm{V} + 1}{2\mathrm{SNR}_\mathrm{H}\mathrm{SNR}_\mathrm{V}\rho^2_\mathrm{HV}} + \frac{0.28(1-\rho^2_\mathrm{HV})^2}{\sigma_{vn}\rho^2_\mathrm{HV}} \right] } , \label{eq:bulk_4} \ee
where $\sigma_{vn}$ is the normalized spectrum width.
Likewise, \citet{Yu2012} studied the spectral polarimetric variables and obtained the analytical expressions for the errors as:
\be \mathrm{bias}(s\hat{Z}_\mathrm{DR}) = \frac{10}{K \ln{10}} \left[ \frac{1+2s\mathrm{SNR}_\mathrm{V}}{s\mathrm{SNR}_\mathrm{V}^2} + (1-s\rho_\mathrm{HV}^2) \right] ,\ee
\be \mathrm{SD}(s\hat{Z}_\mathrm{DR}) = \frac{10}{\sqrt{K} \ln{10}} \sqrt{ \frac{1+2s\mathrm{SNR}_\mathrm{H}}{s\mathrm{SNR}_\mathrm{H}^2} +  \frac{1+2s\mathrm{SNR}_\mathrm{V}}{s\mathrm{SNR}_\mathrm{V}^2} + 2(1-s\rho^2_\mathrm{HV})}  ,\ee
\be \frac{\mathrm{bias}(s\hat{\rho}_\mathrm{HV})}{s\rho_\mathrm{HV}} = \frac{1}{K} \left[ \frac{2s\mathrm{SNR}_\mathrm{H}+3}{8s\mathrm{SNR}_\mathrm{H}^2} + \frac{2s\mathrm{SNR}_\mathrm{V}+3}{8s\mathrm{SNR}_\mathrm{V}^2} + \frac{s\mathrm{SNR}_\mathrm{H} + s\mathrm{SNR}_\mathrm{V} + 1}{4s\mathrm{SNR}_\mathrm{H}s\mathrm{SNR}_\mathrm{V}s\rho^2_\mathrm{HV}} + \frac{(1-s\rho^2_\mathrm{HV})^2}{4s\rho^2_\mathrm{HV}} \right] ,\ee
\be \frac{\mathrm{SD}(s\hat{\rho}_\mathrm{HV})}{s\rho_\mathrm{HV}} = \sqrt{ \frac{1}{K} \left[ \frac{1-2s\mathrm{SNR}_\mathrm{H}}{4s\mathrm{SNR}_\mathrm{H}^2} + \frac{1-2s\mathrm{SNR}_\mathrm{V}}{4s\mathrm{SNR}_\mathrm{V}^2} + \frac{s\mathrm{SNR}_\mathrm{H} + s\mathrm{SNR}_\mathrm{V} + 1}{2s\mathrm{SNR}_\mathrm{H}s\mathrm{SNR}_\mathrm{V}s\rho^2_\mathrm{HV}} + \frac{(1-s\rho^2_\mathrm{HV})^2}{2s\rho^2_\mathrm{HV}} \right] } .\ee
A simple comparison of these expressions shows the differences in variable dependence for bulk and spectral variables. 
For these analyses the SNR in both channels is assumed to be high, and thus, the terms with $\mathrm{SNR}_\mathrm{H,V}$ and $s\mathrm{SNR}_\mathrm{H,V}$ can be neglected.
Consequently, in the absence of noise, the errors do not depend on $Z_\mathrm{DR}$ \citep{Melnikov2004} or $sZ_\mathrm{DR}$, but show a dependence on $\rho_\mathrm{HV}$, $M$ and $\sigma_v$ for bulk variables, and $s\rho_\mathrm{HV}$,  and $K$ for spectral variables.
Although the spectral variables do not explicitly show dependence with $M$ and $\sigma_v$ in the analytical expressions, the thresholded spectral coefficients  are averaged.
Thus, a smaller dependence in the spectral variables is expected because the number of thresholded coefficients depends on $M$ and $\sigma_v$. 
In the next section, the errors of the spectral variables are analyzed for different parameters to illustrate the error behavior of spectral variables in detail.
	
\section{Analysis of the Ideal Estimator}
As described in Chapter \ref{chap_2}, with the use of the periodogram PSD estimator and by averaging independent spectra, DPSDs with better precision can be obtained.
It is expected that the errors will be higher for spectral variables because they mostly depend on $K$, whereas the bulk variables depend on $M$ (generally, $M$ is larger than $K$).
Ideally, the number of independent spectra to obtain estimates with acceptable error levels must be large.
It was determined by \citet{Yu2012} that $K \geq 20$ yields good error levels.
Hereafter, the periodogram DPSD estimator is the one that averages $K$ independent periodogram PSD estimates; and the ideal DPSD estimator becomes the periodogram DPSD estimator with $K=20$. 
It is important to recall that it is not operationally possible to obtain estimates with a large number of independent spectra.
In this section, the errors of the periodogram DPSD estimator for different values of $K$ as functions of relevant parameters are analyzed, so that a standard for comparison for further performance analyses can be established.
The signal parameters that have a potential impact in the quality of the spectral estimates are SNR, $M$, $\sigma_v$, $Z_\mathrm{DR}$, $\rho_\mathrm{HV}$, and $K$.
Since studying the effect of each parameter on the error quality over a wide range of values for all DPSD estimators under consideration would be cumbersome, the analysis of the ideal estimator can help determine the parameters with higher impact on the errors.
Thus, by focusing on these parameters, the complexity of further analyses are greatly simplified.
The signal parameters are summarized in Table \ref{tab:ideal_sim}.
Each point in the error curves are computed using (\ref{eq:error_1})-(\ref{eq:error_6}) with a total of $NK$ independent signals per step in the $x$-axis. 
\begin{table}
\centering
\begin{tabular}{|c|c|}
\hline Simulation parameter&Value\\ \hline \hline
$M$ & 64\\
$N$ & 5000 \\
$\bar{v}_\mathrm{r}$ & $5$ m/s\\
$\sigma_v$ & $2$ m/s \\ 
$\mathrm{SNR}_\mathrm{V}$ & $30$ dB \\
$Z_\mathrm{DR}$ & $1.5$ dB\\
$\rho_\mathrm{HV}$ & 0.90 \\ \hline
$v_\mathrm{a}$ & $15.7$ m/s \\ \hline
\end{tabular}
\caption{Simulation parameters for the analysis of the errors.}
\label{tab:ideal_sim}
\end{table}

\subsection{Dependence with SNR}
The behavior of the errors as a function of the SNR is analyzed using the parameters listed in Table \ref{tab:ideal_sim}, but by varying $\mathrm{SNR}_\mathrm{V}$ between 14 and 30 dB.
In bulk variables, for a fixed $M$, higher SNR yields smaller errors \citep{Melnikov2004}.
In spectral variables, higher SNR will cause more spectral leakage if data windows with insufficient SLL are used. 
When spectral leakage occurs, the effective number of samples above the SNR threshold can unrealistically increase, and it may potentially affect the interpretation of the errors of estimates.
Ideally, for higher SNR, it is expected that the errors are smaller and independent of the SNR values because the thresholded spectral coefficients already have minimal noise contamination.
Additionally, the dependence on SNR for bulk variables is stronger because all $M$ samples are used to compute the estimates and errors, as opposed to using averaged thresholded spectral coefficients.
The bias and standard deviation (SD) of $s\hat{Z}_\mathrm{DR}$ and $s\hat{\rho}_\mathrm{HV}$ as a function of $\mathrm{SNR}_\mathrm{V}$ are shown in Figure \ref{fig:ideal_snr}.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap4/ideal_periodogram_S_K.eps}
\caption{Errors of the spectral polarimetric variables as function of the true V-channel SNR for the periodogram DPSD estimator and different values of $K$. Bias (top left) and SD (bottom left) of $s\hat{Z}_\mathrm{DR}$, normalized bias (top right) and SD (bottom right) of $s\hat{\rho}_\mathrm{HV}$ for $K=2$ (blue), $K=5$ (green), $K=10$ (red), $K=20$ (cyan), and $K=100$ (magenta). Herein, the periodogram DPSD with $K=20$ is considered to be the ``ideal" DPSD estimator.}
\label{fig:ideal_snr}
\end{figure}
Since the noise components in the spectra have $sZ_\mathrm{DR} = 0$ dB and the signal components have $sZ_\mathrm{DR}=1.5$ dB, the errors in $sZ_\mathrm{DR}$ for the cases corresponding to bulk $\mathrm{SNR}_\mathrm{V} < 20$ dB are more biased towards the noise DPSD values, which explains the unusual behavior (smaller biases for smaller SNR).
In other words, the bias of $s\hat{Z}_\mathrm{DR}$ for this estimator is inherently positive and large, and lower SNR values ``help" reduce the bias, though this effect is a function of the bulk $Z_\mathrm{DR}$ chosen.
For example, if a negative $Z_\mathrm{DR}$ is selected, the bias of $sZ_\mathrm{DR}$ will decrease with increasing SNR, as expected.
Likewise, the noise components have $s\rho_\mathrm{HV}=0$, while the signal components have $s\rho_\mathrm{HV}=0.90$; it can be seen that $s\rho_\mathrm{HV}$ is overestimated for lower $K$, while it shows the expected behavior for the curves corresponding to higher $K$.
The bias and SD of $s\hat{Z}_\mathrm{DR}$ and $s\hat{\rho}_\mathrm{HV}$ as a function of $\mathrm{SNR}_\mathrm{V}$ are almost constant for the curves corresponding to $K = 5, 10, 20$ and $100$, for $\mathrm{SNR}_\mathrm{V} \geq 20$.
Clearly, this exercise shows that the estimates with $K=2$ are not reliable even for signals with high SNR; and with higher $K$, the errors become increasingly independent of the SNR (for $\mathrm{SNR} \geq 20$) and gradually smaller.
Higher bulk SNR translates into more spectral coefficients that meet or exceed the SNR threshold ($\mathrm{SNR}_\mathrm{H,V} \geq 20$ dB); and thus, the SD of the estimates is reduced (higher number of averaged samples), while the biases tend to an asymptotic value.
The rest of the analysis is conducted for an SNR of $30$ dB to ensure minimal noise contamination in the spectral coefficients of the signal.

\subsection{Dependence with $M$}
%The signal parameters for this analysis are summarized in Table \ref{tab:ideal_m}.
In bulk variables, the errors have a strong dependence with $M$, while a weaker dependence is expected for spectral variables.
The errors as a function of the number of samples ($M$) are shown for different number of independent spectra ($K$) in Figure \ref{fig:ideal_m}.
The signal parameters are shown in Table \ref{tab:ideal_sim}, except different values of $M$ are analyzed.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap4/ideal_periodogram_M_K.eps}
\caption{Same as Figure \ref{fig:ideal_snr}, but as a function of the number of samples ($M$) for different values of $K$.}
\label{fig:ideal_m}
\end{figure}
As usual, it is desired to have small errors in the estimates, and once again it can be seen how the errors improve for higher $K$.
Increasing $M$ increases the frequency resolution and translates into having more sample points in the spectral estimates.
Although the choice of $M$ does not necessarily affect the accuracy of the DPSD estimates, averaging SNR-thresholded spectral coefficients does depend on $M$.
Logically, by increasing $M$, the number of spectral coefficients above the SNR threshold also increases.
A small dependence of the errors and $M$ can be observed for small $M$, where the number of samples above the SNR threshold is insufficient to produce good estimates, and a bias towards the noise value of $0$ dB can be observed (similar to the case with small SNR).
Furthermore, for $M\geq 32$, the biases are approximately constant, while an improvement in the SDs can be seen.
Hereafter, the number of samples for the analyses is set to $M=64$ to resemble operational settings.

\subsection{Dependence with $\sigma_v$}
The signal parameters used in this analysis are those of Table \ref{tab:ideal_sim}, but for different values of $\sigma_v$ from $0.5$ to $3$ m/s.
In bulk variables, as seen in (\ref{eq:bulk_1})-(\ref{eq:bulk_4}), the errors have a term that depends on the spectrum width; however, spectral variables do not strongly depend on the spectrum width.
As with the previous analysis for $M$, the actual value of $\sigma_v$ does not affect the accuracy of the DPSD estimates, but it does affect the averaging of SNR-thresholded spectral coefficients.
Thus, a small dependence with $\sigma_v$ is expected.
For spectral variables, the dependence of the errors with the spectrum width is similar to that with $M$ (Figure \ref{fig:ideal_w}), though slightly stronger because the number of samples above the threshold increases at a higher rate for increasing $\sigma_v$ than for increasing $M$.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap4/ideal_periodogram_W_K.eps}
\caption{Same as Figure \ref{fig:ideal_snr}, but as a function of the true spectrum width ($\sigma_v$) for different values of $K$.}
\label{fig:ideal_w}
\end{figure}
The biases of the DPSD estimates have stronger dependence with $\sigma_v$ for smaller $K$, as seen in the blue curves corresponding to $K=2$, while for higher $K$, the biases become almost independent of $\sigma_v$.
For all $K$, the SDs decrease with increasing $\sigma_v$ as expected, because of the increased number of samples above the SNR threshold.
The spectrum width for the simulations is selected to match typical raindrop signals with $\sigma_v = 2$ m/s \citep{Fang2004}.

\subsection{Dependence with $Z_\mathrm{DR}$}
As previously mentioned, in the absence of noise, it is expected that the errors do not show a dependence with $Z_\mathrm{DR}$ \citep{Melnikov2004}.
The simulation parameters are listed in Table \ref{tab:ideal_sim} with the $Z_\mathrm{DR}$ varied from $-2.5$ to $2.5$ dB (Figure \ref{fig:ideal_d}).
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap4/ideal_periodogram_D_K.eps}
\caption{Same as Figure \ref{fig:ideal_snr}, but as a function of the true differential reflectivity ($Z_\mathrm{DR}$) for different values of $K$.}
\label{fig:ideal_d}
\end{figure}
For the curves corresponding to $K>2$, it is observed that the errors remain almost constant as a function of $Z_\mathrm{DR}$, as expected.
However, it can be seen that $K=2$ is insufficient to yield good estimates, as the bias in $s\hat{Z}_\mathrm{DR}$ is weakly dependent on the true value of $Z_\mathrm{DR}$.
In other words, the estimates are influenced by the $Z_\mathrm{DR}$ of the noise ($0$ dB).
For further analyses, a $Z_\mathrm{DR}$ of $1.5$ dB is used to resemble slightly larger raindrops of about 2.70 mm \citep{Bringi2001}.

\subsection{Dependence with $\rho_\mathrm{HV}$}
The correlation coefficient is varied from 0.70 to 0.99 to obtain the curves for Figure \ref{fig:ideal_r}.
Both bulk and spectral variables have a similar, strong dependence with $\rho_\mathrm{HV}$.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap4/ideal_periodogram_R_K.eps}
\caption{Same as Figure \ref{fig:ideal_snr}, but as a function of the true correlation coefficient ($\rho_\mathrm{HV}$) for different values of $K$.}
\label{fig:ideal_r}
\end{figure}
It is observed that for higher $\rho_\mathrm{HV}$, the errors become smaller.
Additionally, the dependence with $\rho_\mathrm{HV}$ can be interpreted as a measure of signal consistency, as the estimates for high true $\rho_\mathrm{HV}$ values are less dispersed, both in bias and variance; while the estimates for lower true $\rho_\mathrm{HV}$ values are highly dispersed and inaccurate.
Although most hydrometeors exhibit high correlation coefficient (i.e., $\rho_\mathrm{HV}>0.98$), the value of $\rho_\mathrm{HV} = 0.90$ was used in most simulations (unless otherwise stated) to reveal potential biases in the DPSDs.

\subsection{Dependence with $K$}
So far, it was shown that for spectral variables, errors show weak to no dependence on most parameters, with the exception of $\rho_\mathrm{HV}$, and the remaining parameter to be analyzed is $K$.
The errors as a function of $K$, are presented for different values of $\rho_\mathrm{HV}$ in Figure \ref{fig:ideal_k}.
Again, the simulation values are those shown in Table \ref{tab:ideal_sim}, with $K$ ranging from 2 to 30, and for $\rho_\mathrm{HV}$ values of 0.80, 0.90, 0.95, and 0.99.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{figures/chap4/ideal_periodogram_K_R.eps}
\caption{Errors of the spectral polarimetric variables as function of the number of independent spectra ($K$) of the periodogram DPSD estimator for different values of $\rho_\mathrm{HV}$. Bias (top left) and SD (bottom left) of $s\hat{Z}_\mathrm{DR}$, normalized bias (top right) and SD (bottom right) of $s\hat{\rho}_\mathrm{HV}$ for $\rho_\mathrm{HV}=0.80$ (blue), $\rho_\mathrm{HV}=0.90$ (green), $\rho_\mathrm{HV}=0.95$ (red), and $\rho_\mathrm{HV}=0.99$ (cyan).}
\label{fig:ideal_k}
\end{figure}
For $K>20$, the improvement in the errors is marginal, while for $K<10$ the quality of the estimates can be significantly degraded.

\section{Analysis of the Single-Dwell Estimator for the Single-Signal Case}
It is understood that a large number of independent spectra for DPSD estimation is not feasible in practice.
The Bootstrap DPSD estimator presented in Chapter \ref{chap_3} was developed to overcome this limitation by allowing the estimation of DPSDs from a single dwell ($K=1$), with minimal resolution loss.
Moreover, there are conventional estimators that are also capable of producing DPSD estimates from a single dwell.
In this section, the performance of the conventional and Bootstrap DPSD estimators is studied for a single signal, and the ideal estimator with $K=20$ is selected as a standard for comparison.
Additionally, the analysis focuses on the errors as a function of $\rho_\mathrm{HV}$ as justified by the previous analysis.

For a fair comparison between the different DPSD estimators, the parameters for the conventional methods are selected such that there is minimal frequency resolution loss and spectral smearing.
That is, for Welch's estimator, the segment length is set to $L=M-1$ with maximum overlap; and for Daniell's estimator, a 3-point moving average filter ($p=1$) is used for PSD smoothing.
Bartlett's estimator is not analyzed because its frequency resolution is degraded by at least a factor of 2.
The Bootstrap DPSD estimates are obtained using $K'=20$ pseudo-realizations.
The analysis parameters are listed in Table \ref{tab:ideal_sim}, with $\rho_\mathrm{HV}$ varying from 0.85 to 0.99.
The data window for the analysis is a Blackman-Nuttall window, though it should be noted that other windows with sufficient SLL yield similar results.
%\begin{table*}
%\centering
%\caption{Simulation parameters}
%\begin{tabular}{c|c|c|c|c}
%
%&Single dwell single signal&Multiple dwell single signal&\multicolumn{2}{c}{Single dwell dual signal}\\
%\hline \hline
%Parameter&Value & Value &Raindrops&Debris \\
%\hline
%$\mathrm{SNR}$ & 30 dB &  30 dB & 30 dB & 30 dB \\
%$v_\mathrm{r} $& -6 m/s & -6 m/s & 8 m/s & $\{ -2, -1, \ldots, 8 \}$ m/s\\
%$\sigma_v$ & \{1, 2, 4, 6\} m/s& 2 m/s & 2 m/s & 1 m/s \\
%$Z_\mathrm{DR} $& \{-1.5, 0, 1.5\} dB& 1.5 dB & $U(0, 1.5)$ dB & $U(-1.5, 0.5)$ dB \\
%$\rho_\mathrm{HV} $& $\{ 0.5, 0.6, \ldots, 0.9, 0.99 \}$& $\{ 0.7, 0.8, 0.9 \}$ & $U(0.99,1.00)$ &  $U(0.80,0.95)$ \\
%$\phi_\mathrm{DP} $& 0 $^\circ$& 0 $^\circ$ &  0 $^\circ$&  0 $^\circ$\\
%\hline
%\end{tabular} 
%\label{table:1}
%\end{table*}
The performance of the estimators in terms of the statistical errors as a function of the true correlation coefficient is shown in Figure \ref{fig:SDSS}. 
\begin{figure}
\centering
\includegraphics[width=1.00\linewidth]{figures/chap4/analysis_sssd.eps}
\caption{Errors of the spectral polarimetric variables as function of the true $\rho_\mathrm{HV}$ values. Bias (top left) and SD (bottom left) of $s\hat{Z}_\mathrm{DR}$, normalized bias (top right) and SD (bottom right) of $s\hat{\rho}_\mathrm{HV}$ for $Z_\mathrm{DR} =1.5$ dB. The Bootstrap DPSD (blue), Daniell (green), Welch (red), and ideal with $K=20$ (black) estimates are compared.}
\label{fig:SDSS}
\end{figure}
The errors for Welch estimates (red) with the best possible frequency resolution are very high and useless for practical purposes.
It can be observed that the Bootstrap DPSD estimates (blue) are better than Daniell estimates (red) for all cases.
For $s\hat{Z}_\mathrm{DR}$, and $\rho_\mathrm{HV} = 0.90$, the Daniell estimator has a bias of $0.708$ dB, while the biases of the Bootstrap estimator and the ideal estimator are of $0.057$ dB and $0.029$ dB, respectively.
The SDs are $1.187$, $0.686$, and $0.1662$ dB, for the Daniell, Bootstrap, and ideal estimators, respectively.
For the same $\rho_\mathrm{HV} = 0.90$, the normalized biases of $s\hat{\rho}_\mathrm{HV}$ are $0.0344$, $0.0177$, and $0.0006$; and the normalized SDs are $0.0476$, $0.0377$, and $0.0089$, for the Daniell, Bootstrap, and ideal estimators, respectively.
A clear improvement in the quality of the DPSD estimates can be seen for the Bootstrap estimator over conventional methods, especially for the $s\hat{Z}_\mathrm{DR}$ bias.
It is important to note that for higher $\rho_\mathrm{HV}$, the errors of the Bootstrap estimates are closer to the error levels of the ideal estimator.
However, for less coherent signals (i.e., lower $\rho_\mathrm{HV}$), the quality of the estimates is degraded.
Therefore, in practice, with single-dwell DPSD estimates using the Bootstrap DPSD estimator, a good qualitative analysis can be performed but the error levels may not be sufficient for a reliable quantitative analysis.
Nonetheless, the following section will show that the errors can be improved by using multiple dwells.

\section{Analysis of the Multiple-Dwell Estimator for the Single-Signal Case}
The results of the conventional and Bootstrap DPSD estimators using a single dwell were shown in the previous section.
However, it is possible to obtain better error levels by averaging multiple spectra for DPSD estimation (i.e., $K\geq2$) with the estimators under analysis.
As mentioned in Chapter \ref{chap_2}, more spectra can be obtained from different range gates, azimuths, or dwells, and it is important to keep the averaging in any of these dimensions to a minimum in order to avoid degrading the resolution if the results.
For this analysis, the statistical errors of conventional and Bootstrap DPSD estimators for $K\geq2$ are compared to the ideal estimator as depicted in Figure \ref{fig:MDSS} (same simulation parameters as before).
\begin{figure}
\centering
\includegraphics[width=1.00\linewidth]{figures/chap4/analysis_ssmd.eps}
\caption{Errors of the spectral polarimetric variables as functions of the number of independent dwells $K$ for $\rho_\mathrm{HV}=0.90$. Bias (top left) and SD (bottom left) of $s\hat{Z}_\mathrm{DR}$, normalized bias (top right) and SD (bottom right) of $s\hat{\rho}_\mathrm{HV}$. The Bootstrap (blue), Daniell (green), Welch (red), and ideal (black) estimators are shown for comparison.}
\label{fig:MDSS}
\end{figure}
For $\rho_\mathrm{HV}=0.90$, the biases of the ideal estimator are equivalent to the Daniell estimator with $K\simeq 16$, and the Bootstrap estimator with $K\simeq 3$.
The Bootstrap DPSD estimator shows a significant improvement in the biases and in the reduction of the number of independent spectra needed.
However, to meet the SD of the ideal estimates, a considerable number of independent spectra is still needed.
The SDs for the ideal estimator are equivalent to the Daniell estimator with $K\simeq 20$, and the Bootstrap estimator with $K\simeq 9$.
For the Daniell estimator, a marginal improvement is observed, while the Bootstrap estimator shows overall better performance.
When $K>20$, the Welch estimator converges with the ideal estimator, while the Daniell estimator performs marginally better, and the Bootstrap estimator exceeds the performance of the ideal estimator.
One important drawback of the Bootstrap DPSD estimator is the inherent loss of coherence associated with the application of correction strategies, depicted in the negative biases of $s\hat{\rho}_\mathrm{HV}$ for larger $K$.
Still, the normalized bias of $s\hat{\rho}_\mathrm{HV}$ is within 0.002 for $K \geq 2$ and for $\rho_\mathrm{HV} = 0.90$, which is within the error level recommended by \citet{Melnikov2004}.
While this limitation is noted, the Bootstrap DPSD estimator can generally achieve better quality estimates (for a given number of independent spectra) than other estimators.
A more in-depth analysis of the impact of this effect is beyond the scope of this work.
Additionally, it should be indicated that while the spectra in this analysis are independent, in practice, the spectra could be correlated depending on where the additional spectra come from (e.g., different scans vs. adjacent range gates). %and the error improvement might not be as significant as depicted in the error statistics.

\section{Analysis of the Single-Dwell Estimator for the Dual-Signal Case}
\label{sec:dualsignal}
\subsection{Signal Interaction}
In order to discriminate signals in the spectra, the interaction of different signals in a composite weather radar I/Q signal must be understood.
A signal becomes dominant in the spectrum whenever its power relative to the powers of other signals is large (e.g., $>20$ dB), and for these spectral coefficients, the characteristics in the DPSDs become predominantly those of the stronger signal.
The single-signal case is essentially composed of two spectral signatures: the simulated signal and the white noise.
As such, the spectra in the single-signal analysis can be separated into three regions: dominant signal, dominant noise, and the transition where neither is dominant.
That is, if the PSD can be represented as:
\be sS(k) = sS_0(k) + sS_1(k)  , \label{eq:spectral_splusn_1} \ee
where $sS_1$ is the PSD of the signal without noise, and $sS_0$ is the PSD of noise; 
then the DPSDs become:
\be sZ_\mathrm{dr}(k) \simeq \frac{sS_0(k) sZ_{\mathrm{dr},0}(k) + sS_1(k) sZ_{\mathrm{dr},1}(k)}{sS(k)} , \ee
and
\be s\rho_\mathrm{HV}(k) \simeq \frac{sS_0(k) s\rho_{\mathrm{HV},0}(k) + sS_1(k) s\rho_{\mathrm{HV},1}(k)}{sS(k)} . \label{eq:spectral_splusn_3} \ee
Based on this, when $sS_1\gg sS_0$, the signal is dominant and $sZ_\mathrm{dr} \simeq sZ_{\mathrm{dr},1}$, $s\rho_\mathrm{HV} \simeq s\rho_{\mathrm{HV},1}$; and when $sS_1\ll sS_0$, noise is dominant and $sZ_\mathrm{dr} \simeq sZ_{\mathrm{dr},0}$, $s\rho_\mathrm{HV} \simeq s\rho_{\mathrm{HV},0}$.
For spectral coefficients where neither the signal nor the noise is dominant, the DPSDs are a weighted average of the DPSDs of the individual signals; i.e., they are neither the DPSDs of the signal nor the noise.

A similar behavior is expected when an additional signal is present in the spectra.
The purpose of this dual-signal analysis is to study the effects of having an additional signal in the spectra, both in terms of statistical errors and in terms of the ability to discriminate the different signals.
With two signals and noise, (\ref{eq:spectral_splusn_1})-(\ref{eq:spectral_splusn_3}) become:
\be sS(k) = sS_0(k) + sS_1(k) + sS_2(k) , \ee
where $sS_2$ is the PSD of the additional signal; 
and the DPSDs become:
\be sZ_\mathrm{dr}(k) \simeq \frac{sS_0(k) sZ_{\mathrm{dr},0}(k) + sS_1(k) sZ_{\mathrm{dr},1}(k) + sS_2(k) sZ_{\mathrm{dr},2}(k) }{sS(k)} , \ee
\be s\rho_\mathrm{HV}(k) \simeq \frac{sS_0(k) s\rho_{\mathrm{HV},0}(k) + sS_1(k) s\rho_{\mathrm{HV},1}(k) + sS_2(k) s\rho_{\mathrm{HV},2}(k)}{sS(k)} .  \ee
To demonstrate this, the ideal DPSDs for each signal are estimated individually and the power-weighted average of the DPSDs is then computed.
This is compared with the ideal DPSDs estimated from the composite I/Q signal, as shown in Figure \ref{fig:ds_interact}, for a unimodal and a bimodal case.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.29\linewidth}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|c|c|c|}
\hline Parameter & Signal 1& Signal 2 \\ \hline 
$\mathrm{SNR}$ & $20$ dB& 40 dB \\
$\hat{v}_\mathrm{r}$ & $0$ m/s & $0$ m/s \\
$\sigma_v$ & $3.5$ m/s & $1$ m/s  \\
$Z_\mathrm{DR}$ & $1.5$ dB & $-1.5$ dB \\
$\rho_\mathrm{HV}$ & $0.99$ & $0.90$ \\ \hline
\end{tabular}%
}
\end{minipage}
\begin{minipage}[c]{0.7\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_ds_interact.eps}
\end{minipage}
\\(a)\\
\begin{minipage}[c]{0.29\linewidth}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|c|c|c|}
\hline Parameter & Signal 1& Signal 2 \\ \hline 
$\mathrm{SNR}$ & $20$ dB& 35 dB \\
$\hat{v}_\mathrm{r}$ & $-3$ m/s & $3$ m/s \\
$\sigma_v$ & $2.5$ m/s & $0.75$ m/s  \\
$Z_\mathrm{DR}$ & $0.5$ dB & $2$ dB \\
$\rho_\mathrm{HV}$ & $0.99$ & $0.85$ \\ \hline
\end{tabular}%
}
\end{minipage}
\begin{minipage}[c]{0.7\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_ds_interact_2.eps}
\end{minipage}
\\(b)
\caption{Dual-signal spectral coefficient interaction and composite effect on the DPSD for (a) unimodal case, and (b) bimodal case. The mean of the ideal estimates of signal 1 (blue) and signal 2 (green) are computed separately. The mean ideal estimate of the composite signal (continuous black) is compared to the power-weighted average of the DPSDs of signals 1 and 2 (dashed black).}
\label{fig:ds_interact}
\end{center}
\end{figure}
The blue and green lines are the averaged ideal estimates as obtained from the individual I/Q signals.
The continuous black line represents the averaged ideal estimate of the composite signal (i.e., the I/Q signal composed of signals 1 and 2 plus noise).
The dashed black line is computed as the power-weighted average of the DPSDs of signals 1 and 2 (the noise is negligible in this case).
When the two signal powers are comparable, the DPSDs do not exactly follow one signal or the other.
Instead, they show characteristics that fall in between the two signals, and as the power of one signal increases, it gradually becomes dominant.
The unimodal case shows how two signals may appear as one, even though the spectrum width is exaggerated for clarity.
Here, the DPSDs provide evidence that a narrower signal is overlaid on top of a wider signal, and that these two have different polarimetric characteristics.
In the bimodal case, the two different signals show a difference in velocity, which makes them somewhat easier to identify.
Once again, both signals show different polarimetric characteristics which can be used to classify them accordingly.
While the power-weighted average of the DPSDs does not exactly follow the composite signal's DPSDs for regions with comparable SNR, both curves are similar for higher SNR.
Thus, qualitatively, it can be seen assumed that the DPSDs are a power-weighted average of the DPSDs of each signal.
Moreover, this suggests the possibility for discriminating signals that are overlapped by identifying distinct polarimetric signatures in the spectra.
A few representative examples will be presented next.

\subsection{Case Studies}
There are various signal parameters that affect the ability to separate different spectral signatures; some of them are the difference between the mean radial velocities, the spectrum widths, the difference between the SNRs, and the polarimetric characteristics.
The parameters for the signals used hereafter are summarized in Table \ref{tab:ds_cases}, with $N=1000$ iterations.
\begin{table}
\centering
\footnotesize
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{3}{*}{Case} & \multicolumn{5}{|c|}{Signal 1} & \multicolumn{5}{|c|}{Signal 2} \\ \cline{2-11}
& SNR & $\bar{v}_\mathrm{r}$ & $\sigma_v$ & $Z_\mathrm{DR}$ & $\rho_\mathrm{HV}$ & SNR & $\bar{v}_\mathrm{r}$ & $\sigma_v$ & $Z_\mathrm{DR}$ & $\rho_\mathrm{HV}$ \\
&(dB)&(m/s)&(m/s)&(dB)&&(dB)&(m/s)&(m/s)&(dB)&\\ \hline \hline
0&25&-6&2&1.5&0.995&40&9&1&-1.5&0.97\\ \hline
1&\multirow{4}{*}{25}&\multirow{4}{*}{-6}&\multirow{4}{*}{2}&\multirow{4}{*}{1.5}&\multirow{4}{*}{0.995}&\multirow{4}{*}{40}&-6&\multirow{4}{*}{1}&\multirow{4}{*}{-1.5}&\multirow{4}{*}{0.97}\\
2&&&&&&&-4&&&\\
3&&&&&&&-2&&&\\
4&&&&&&&0&&&\\ \hline
5&\multirow{4}{*}{25}&\multirow{4}{*}{-6}&\multirow{4}{*}{2}&\multirow{4}{*}{1.5}&\multirow{4}{*}{0.995}&\multirow{4}{*}{25}&-6&\multirow{4}{*}{1}&\multirow{4}{*}{-1.5}&\multirow{4}{*}{0.97}\\
6&&&&&&&-4&&&\\
7&&&&&&&-2&&&\\
8&&&&&&&0&&&\\ \hline
9&\multirow{4}{*}{40}&\multirow{4}{*}{-6}&\multirow{4}{*}{2}&\multirow{4}{*}{1.5}&\multirow{4}{*}{0.995}&\multirow{4}{*}{25}&-6&\multirow{4}{*}{1}&\multirow{4}{*}{-1.5}&\multirow{4}{*}{0.97}\\
10&&&&&&&-4&&&\\
11&&&&&&&-2&&&\\
12&&&&&&&0&&&\\ \hline
\end{tabular}
\caption{Signal parameters for dual-signal analysis cases.}
\label{tab:ds_cases}
\end{table}
Signal 1 is assumed to be a highly coherent signal with values resembling those of typical raindrops, while signal 2 is less coherent with negative $Z_\mathrm{DR}$ resembling hypothetical tornadic debris.
Although a Gaussian assumption for debris signals is not realistic, it serves to illustrate the presence of signals with different polarimetric characteristics in the spectra, as will be shown in the following cases.
\subsubsection{Case 0}
When the spectra of the signals do not overlap (i.e., the difference in mean radial velocities is sufficiently large and the spectrum widths are not too wide), the signals can be easily separated by their velocities, and by assessing their spectral polarimetric variables, it may be possible to classify them accordingly (Figure \ref{fig:ds_dh_no}).
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\scriptsize
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o0.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\scriptsize
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h0.png}
\\(b)
\end{minipage}
\caption{Dual-signal analysis, case 0. (a) The mean of the DPSD estimates of Bootstrap (blue), Daniell (green), and ideal (black) estimates, and the true bulk polarimetric variables of signal 1 (dashed red) and signal 2 (dashed dark red) are shown. (b) A 2D histogram of $s\hat{\rho}_\mathrm{HV}$ and $s\hat{Z}_\mathrm{DR}$ for thresholded spectral coefficients in logarithmic ($\log_{10}$) scale, of Bootstrap (top), Daniell (middle), and ideal (bottom) estimates, with $\times$ indicating the bulk estimates of the composite (black) and individual (red) signals.}
\label{fig:ds_dh_no}
\end{center}
\end{figure}
The average of $N=1000$ DPSD estimates of the signals is shown in Figure \ref{fig:ds_dh_no}a, for the Bootstrap ($K=1$), Daniell ($K=1$), and ideal ($K=20$) estimators.
Additionally, the values for the true bulk polarimetric variables for the signals are plotted in dashed lines.
Whenever the difference in radial velocity is large enough, there is no ambiguity in determining the presence of different signals, and with the DPSDs, the polarimetric characteristics of the signals can be obtained.
In Figure \ref{fig:ds_dh_no}b, a 2D histogram (hereafter, histogram) of $s\hat{\rho}_\mathrm{HV}$ and $s\hat{Z}_\mathrm{DR}$ is computed for the thresholded spectral coefficients, over $N=1000$ iterations, in logarithmic ($\log_{10}$) scale.
The red marker indicates the true values of the signals, while the black marker indicates the bulk estimate of the composite signal.
For this and upcoming cases, it is important to note that the distributions are skewed towards high $s\rho_\mathrm{HV}$, but the mean values depicted in the DPSD estimates (e.g., in Figure \ref{fig:ds_dh_no}a) are in fact closer to the markers (e.g., in Figure \ref{fig:ds_dh_no}b) than it appears.
The logarithmic scale was chosen to emphasize the distributions of the polarimetric characteristics of the SNR-thresholded spectral coefficients, since the mean values of the DPSD estimates may not always correctly represent important differences between the estimators under analysis.
Comparing the estimators, the effect of increasing $K$ in the ideal estimator can be seen to reduce the errors.
Additionally, in the Bootstrap estimates, the difference between the values of the signals is clearer than with the Daniell estimates, showing a better performance for the Bootstrap DPSD estimator.
While the Bootstrap estimates are slightly more dispersed when compared to the ideal estimates, it should be noted that the Bootstrap estimates are obtained using $K=1$, while the ideal estimates are obtained with $K=20$, which is a significant difference.
The actual values of the SNR, the mean radial velocities, or the spectrum widths of the signals are irrelevant in this case (assuming the signals are not ``leaking" into one another), since the separation between the signals is evident. 
The error statistics for each individual signal are equivalent to those of the single-signal analysis.

The spectrum widths and the difference between the mean radial velocities of the signals determine the degree to which the spectral coefficients of one signal are ``leaking" into the spectral coefficients of other signals.
Now that it is better understood how different signals interact to yield the DPSDs, it can be inferred from the SNR which spectral coefficients are more relevant.
Additionally, certain patterns in $s\hat{Z}_\mathrm{DR}$ and $s\hat{\rho}_\mathrm{HV}$ can aid in determining characteristics of dominant (or non-dominant) signals in the spectra.
In practice, however, it would be impossible to determine the amount of overlap to perform a one-to-one classification of the polarimetric variables for each spectral coefficient.
Nevertheless, even if the presence of two or more different signals is not readily apparent in the spectra, it is possible to obtain a meaningful qualitative inference of the underlying phenomena with the DPSDs.

\subsubsection{Cases 1-4}
For this set of cases, the hypothetical debris signal is assumed to be higher in power.
The mean radial velocity of the debris signal is varied such that the difference in radial velocity with the hypothetical raindrop signal is $0$ for case 1, $\sigma_{v,1}$ for case 2, 2$\sigma_{v,1}$ for case 3, and 3$\sigma_{v,1}$ for case 4.
Case 1 (Figure \ref{fig:ds_dh1}a) shows a complete overlap of the debris spectral signature on the raindrop spectral signature, which yields a unimodal PSD.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o1.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h1.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 1.}
\label{fig:ds_dh1}
\end{center}
\end{figure}
An analysis based on the PSD alone would not provide any useful information about the additional signal.
However, the spectral signatures of $Z_\mathrm{DR}$ and $\rho_\mathrm{HV}$ show evidence of different polarimetric characteristics.
Spectral techniques to extract information from the DPSDs could be developed for various applications, though this is beyond the scope of this work.
In Figure \ref{fig:ds_dh1}b, the histograms of the ideal estimates show two distinct regions distributed near the true values of the signals.
The Bootstrap estimates are also able to produce these two regions, though the errors are visibly higher than the ideal estimates (higher bias and variance).
With the Daniell estimates, the values are distributed across a wider range of $Z_\mathrm{DR}$ and mostly high $\rho_\mathrm{HV}$ values.
The distinction between the values of the two different signals is not apparent, and thus it fails to produce distinctive signatures in the DPSDs.

The subsequent cases show bimodal spectra (Figures \ref{fig:ds_dh2}a-\ref{fig:ds_dh4}a), with the difference in mean radial velocities increased to $\sigma_{v,1}$, $2\sigma_{v,1}$, and $3\sigma_{v,1}$ for cases 2, 3, and 4, respectively.
With the increasing separation between the signals' mean radial velocities, it becomes easier to identify the different spectral signatures based on the PSD.
With the aid of the DPSDs, the influence of the debris signal can be detected and the raindrop signal can be extracted to obtain more accurate measurements of its properties.
Additionally, the histograms (Figures \ref{fig:ds_dh2}b-\ref{fig:ds_dh4}b) of the Bootstrap and ideal estimates indicate the presence of two distinct signals, with the separation becoming clearer with increasing difference in the mean radial velocities.
The Daniell estimates, while they show a weak separation for higher $s\hat{\rho}_\mathrm{HV}$ values, are not able to properly capture the less coherent signal.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o2.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h2.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 2.}
\label{fig:ds_dh2}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o3.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h3.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 3.}
\label{fig:ds_dh3}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o4.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h4.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 4.}
\label{fig:ds_dh4}
\end{center}
\end{figure}

\subsubsection{Cases 5-8}
This set of cases assume the raindrop and the debris signals are comparable in power.
Similar to the previous set, the mean radial velocity of the debris signal is varied such that the difference in radial velocities is: $0$ for case 5, $\sigma_{v,1}$ for case 6, 2$\sigma_{v,1}$ for case 7, and 3$\sigma_{v,1}$ for case 8.
A unimodal spectrum is observed for case 5 (Figure \ref{fig:ds_dh5}a).
Here, since the signals are comparable in power, neither signal is dominant across the width of the debris signal.
As such, the spectral polarimetric variables show $s\hat{Z}_\mathrm{DR}$ values in between $Z_{\mathrm{DR},1}$ and $Z_{\mathrm{DR},2}$, while $s\hat{\rho}_\mathrm{HV}$ is slightly less coherent due to the mixing of signals of different nature.
Outside this range, the raindrop signal, which is wider, becomes slightly more dominant and the DPSDs tend to its true bulk polarimetric variable values.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o5.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h5.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 5.}
\label{fig:ds_dh5}
\end{center}
\end{figure}
Although it is not possible to quantitatively retrieve accurate estimates of the debris signal, the raindrop signal may be reconstructed based on the most accurate information available.
Thus, it would be possible to obtain more accurate estimates of the raindrop signal properties; e.g., for retrieval of bulk polarimetric estimates.
While the histogram of the ideal estimator (Figure \ref{fig:ds_dh5}b, bottom) shows the values spread close to the true values, the debris signal is not dominant for any spectral coefficients and the likely value of $s\hat{Z}_{\mathrm{DR},2}$ and $s\hat{\rho}_{\mathrm{HV},2}$ is shifted towards $Z_{\mathrm{DR},1}$ and $\rho_{\mathrm{HV},1}$.
Likewise, the Bootstrap histogram (Figure \ref{fig:ds_dh5}b, top) does not show a clear separation of two signals, but the presence of the dominant signal given by $s\hat{Z}_\mathrm{DR} \approx s\hat{Z}_{\mathrm{DR},1}$, and $s\hat{\rho}_\mathrm{HV} \approx s\hat{\rho}_{\mathrm{HV},1}$ is evident.
Additionally, the Daniell histogram values (Figure \ref{fig:ds_dh5}b, middle) are inaccurate even to determine the properties of the dominant signal.

With a mean radial velocity difference of $\sigma_{v,1}$ (case 6, Figure \ref{fig:ds_dh6}), a minimal discrimination improvement in the histograms can be seen, though a clear separation of the signals is still not detectable for the Bootstrap estimates.
Additionally, the PSD is still somewhat unimodal, but the DPSDs show evidence of different polarimetric characteristics in the spectra.
The previous analysis for case 5 still holds true.
It is possible to filter out the non-dominant signal to obtain a better estimate of the dominant signal (which, in this case, is the raindrop signal).
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o6.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h6.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 6.}
\label{fig:ds_dh6}
\end{center}
\end{figure}

In cases 7 and 8, the difference in mean radial velocities are $2\sigma_{v,1}$ and $3\sigma_{v,1}$, respectively (Figures \ref{fig:ds_dh7} and \ref{fig:ds_dh8}).
The bimodal spectra become more evident and the histograms of the Bootstrap and ideal estimates show a separation closer to the actual values of the signals.
Even the Daniell estimates show a weak separation in the $s\hat{Z}_\mathrm{DR}$ range, but they are unable to accurately represent the $s\hat{\rho}_\mathrm{HV}$ of less coherent signals, as previously mentioned.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o7.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h7.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 7.}
\label{fig:ds_dh7}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o8.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h8.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 8.}
\label{fig:ds_dh8}
\end{center}
\end{figure}

\subsubsection{Cases 9-12}
In these cases, it is assumed the raindrop signal has higher power.
For cases 9 and 10 (Figures \ref{fig:ds_dh9} and \ref{fig:ds_dh10}), the PSDs are unimodal because the raindrop signal, which is wider and stronger, masks the debris signal.
The information about the debris signal is completely obscured in the histograms, and even the ideal estimator fails in these cases.
There are subtle signatures in the DPSDs that another signal is obscured by this stronger signal; they are characterized by a slight loss in $s\hat{\rho}_\mathrm{HV}$ and a variation in $s\hat{Z}_\mathrm{DR}$ towards the true value of the debris signal.
Despite this, any meaningful qualitative information about the masked signal is lost.
In a practical weather radar situation, if a hydrometeor signal is completely dominant, that would not constitute a problem.
However, if the dominant signal is of non-hydrometeor nature, the weather information would be irretrievable.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o9.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h9.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 9.}
\label{fig:ds_dh9}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o10.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h10.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 10.}
\label{fig:ds_dh10}
\end{center}
\end{figure}

For case 11 (Figure \ref{fig:ds_dh11}), the signals are simulated with a difference in mean radial velocities of $2\sigma_{v,1}$, but with the raindrop signal being higher in power, the PSD still appears to be unimodal.
However, the DPSDs show a more noticeable variation as the raindrop signal weakens and the influence of debris signal becomes stronger.
The histogram of the ideal estimates show a distribution around the true value of the dominant signal, and also around the weighted values where neither signal is dominant.
As previously mentioned, this shifted distribution can be attributed to the presence of a different signal, albeit not quantitatively meaningful due to the poor accuracy.
The performance of the Bootstrap estimates in detecting this additional signal is inferior as the distribution in the histogram is mostly around the true value of the raindrop signal.
Nevertheless, with the information provided by the DPSDs, it is possible to filter out spectral coefficients with different polarimetric characteristics and focus on the dominant characteristics; e.g., to get better bulk estimates of the polarimetric variables.
Once again, Daniell estimates have high variance on $s\hat{Z}_\mathrm{DR}$ and are skewed towards higher $s\hat{\rho}_\mathrm{HV}$, and thus, are unreliable.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o11.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h11.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 11.}
\label{fig:ds_dh11}
\end{center}
\end{figure}

The mean radial velocity difference is set to $3\sigma_{v,1}$ for Case 12 (Figure \ref{fig:ds_dh12}).
Note that the signals are sufficiently apart in velocity such that the PSD is somewhat bimodal and the DPSDs are closer to their respective true values.
This can be detected in the histograms as the values are more closely distributed around the true values of the signals.
While both the ideal and the Bootstrap estimates show two distinct signals (with the Bootstrap estimates being less accurate), the Daniell estimates are still inferior to the Bootstrap estimates.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_o12.eps}
\\(a)
\end{minipage}
\begin{minipage}[c]{0.68\linewidth}
\centering
\includegraphics[width=1\linewidth]{figures/chap4/analysis_h12.png}
\\(b)
\end{minipage}
\caption{Same as Figure \ref{fig:ds_dh_no}, but for case 12.}
\label{fig:ds_dh12}
\end{center}
\end{figure}

\subsubsection{Recap of All Cases}
It was clear from the previous analysis that the Daniell estimates performed the worst, mainly due to the inability to correctly estimate the DPSDs of less coherent signals (the $s\hat{\rho}_\mathrm{HV}$ distributions were particularly skewed).
As summarized in Table \ref{tab:recap}, in 8 out of the 13 cases under analysis, the histogram of Bootstrap estimates showed distinct distributions near the true $Z_\mathrm{DR}$ values of the signals, though with a somewhat skewed distribution in $s\hat{\rho}_\mathrm{HV}$, with a mean bias similar to the values from the single signal single dwell analysis.
\begin{table}
\centering
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|}
\cline{2-6} \multicolumn{1}{c}{}		&  \multicolumn{5}{|c|}{Cases} \\ \hline
\multirow{2}{*}{Methods}	& 1, 5, 9 & 2, 6, 10 	& 3, 7, 11 	& 4, 8, 12 & 0  \\ 
 & ($|\Delta \bar{v}_r| = 0$) & ($|\Delta \bar{v}_r| = \sigma_{v,1}$) & ($|\Delta \bar{v}_r| = 2\sigma_{v,1}$) & ($|\Delta \bar{v}_r| = 3 \sigma_{v,1}$) & ($|\Delta \bar{v}_r| > 3 \sigma_{v,1}$) \\ \hline
Daniell	& None& None& 1 out of 3 ($sZ_\mathrm{DR}$ only)& 1 out of 3 ($sZ_\mathrm{DR}$ only)& All\\
Bootstrap	& 1 out of 3& 1 out of 3& 2 out of 3& All& All\\
Ideal		& 2 out of 3& 2 out of 3& All& All & All\\
\hline
\end{tabular}
\caption{Ability to distinguish signal constituents.}
\label{tab:recap}
\end{table}
A successful separation in the histogram means that the polarimetric characteristics of two distinct signals were properly identified for these cases (in a qualitative manner).
Likewise, the ideal estimates were slightly more successful, with 11 out of 13 cases, and obviously with less errors.
However, it should be emphasized that the ideal estimates used $K=20$ independent spectra, whereas Daniell and Bootstrap estimates were obtained using only one independent spectrum ($K=1$).
Additionally, the cases included unimodal and bimodal spectra.
In the three cases where the Bootstrap estimates performed poorly, it was shown that there was evidence of dispersion in the distribution.
In these situations, techniques such as filtering of non-desired signals could be used to further improve the estimation of the properties of the desired signal.
Furthermore, in two cases where a wider and stronger signal was completely dominating the weaker signal, not even the ideal estimates were able to identify the distinct spectral signatures.
It should be noted that there are many cases in which the identification may not be at all possible even with an ideal DPSD estimator; e.g., wide high-power signal completely masking a narrow weaker signal, signals of similar widths and mean radial velocities, signals with similar polarimetric characteristics and mean radial velocities, etc.

\section{Summary}
%The methodology for this chapter was introduced, which included how the signals for the analyses are generated, how the statistical errors are computed to assess the performance of the different estimators under analysis, and how the bulk variables are related to the spectral variables in terms of estimation and estimate errors.
%With an in-depth analysis of the dependence of the ideal estimator with different parameters, the complexity of the analysis is simplified by selecting the most influential parameters in the error statistics.
%Consequently, a single signal single dwell analysis is presented, showing 
The Bootstrap DPSD estimator shows superior error statistics when compared to conventional DPSD estimators under similar conditions (i.e., minimal frequency resolution loss).
Moreover, it allows estimation of DPSDs with better quality using a single dwell ($K=1$), without resorting to averaging PSDs in time, range, azimuth or frequency.
However, if it is acceptable to compromise the resolution in any of those dimensions, the Bootstrap DPSD estimates with just a few dwells produce accurate results with better performance than the conventional DPSD estimators.
In such cases, the bias levels of the ideal DPSD estimator ($K=20$) are equivalent to the Bootstrap DPSD estimator with $K=5$; and the variance levels are equivalent to the Bootstrap DPSD estimator with $K=13$.
These numbers show a significant reduction in the number of independent spectra needed to achieve desirable error levels (and to consequently produce accurate quantitative spectral analyses), though they may still be somewhat elevated to be operationally feasible. 
The dual-signal analysis showed that the single-dwell Bootstrap DPSD estimates still yield useful results for meaningful qualitative spectral analysis, and is  superior than the conventional DPSD estimator.
For bimodal (or multimodal) spectra, the Bootstrap DPSD estimates are able to identify distinct spectral signatures, regardless of the difference in mean radial velocities, as long as the polarimetric characteristics of the signals are not too similar.
Additionally, even with certain unimodal spectra composed of different signals, the Bootstrap DPSD estimator is able to identify distinct spectral features, though not as good as (and with more errors than) the ideal estimates.
The next chapter discusses the results of applying the Bootstrap DPSD estimator on data from a simulated environment and from real observations. 